Microstructures and anti-phase boundaries in long-range lattice systems

TL;DR

This paper investigates how long-range convex interactions influence the microstructure and anti-phase boundaries in one-dimensional lattice systems, revealing universal patterns of minimizers driven by interfacial energy.

Contribution

It demonstrates that long-range interactions lead to universal, M-periodic minimizers in non-convex lattice systems, extending understanding of microstructure formation.

Findings

M-periodic minimizers are generated by competition between short-range oscillations and long-range order.

The shape of minimizers is universal and independent of energy details.

Number and shape of minimizers increase as the interaction range M diverges.

Abstract

We study the effect of long-range interactions in non-convex one-dimensional lattice systems in the simplified yet meaningful assumption that the relevant long-range interactions are between $M$-neighbours for some $M\ge 2$ and are convex. If short-range interactions are non-convex we then have a competition between short-range oscillations and long-range ordering. In the case of a double-well nearest-neighbour potential, thanks to a recent result by Braides, Causin, Solci and Truskinovsky, we are able to show that such a competition generates $M$-periodic minimizers whose arrangements are driven by an interfacial energy. Given $M$, the shape of such minimizers is universal, and independent of the details of the energies, but the number and shapes of such minimizers increases as $M$ diverges.